Beat modulation

Figure 2.26. Quantum beat-modulated fluorescence decay and its Fourier transform for the l W level of Sj thiophosgene. (Reprinted with permission from Ref. [42].)...

Fourth, the Si —> S0 fluorescence exhibits loss of emission at low Si excess vibrational energy and reappearance at very large excess energies, consistent with predissociation. A quantum beat-modulated Si fluorescence decay was observed at energies corresponding to the expected position of the T2 (71,71 ) state. 

At 1400 cm of excess energy in anthracene, we observed quantum beats in the dispersed fluorescence with a large modulation depth. The beats modulation depth is sensitive to the excess energy and to the fluorescence detection wavelength. 

The second major section (Section III), comprising the bulk of the chapter, pertains to the studies of IVR from this laboratory, studies utilizing either time- and frequency-resolved fluorescence or picosecond pump-probe methods. Specifically, the interest is to review (1) the theoretical picture of IVR as a quantum coherence effect that can be manifest in time-resolved fluorescence as quantum beat modulated decays, (2) the principal picosecond-beam experimental results on IVR and how they fit (or do not fit) the theoretical picture, (3) conclusions that emerge from the experimental results pertaining to the characteristics of IVR (e.g., time scales, coupling matrix elements, coupling selectivity), in a number of systems, and (4) experimental and theoretical work on the influence of molecular rotations in time-resolved studies of IVR. Finally, in Section IV we provide some concluding remarks. 

The situation changed, however, with two advances. The first advance was the discovery that in the S, - S0 spectrum of jet-cooled anthracene a second band exists (at S, + 1420 cm-1), the excitation of which gives rise to quantum beat-modulated fluorescence decays.40 Besides indicating a somewhat more global importance to the beat phenomenon in anthracene, the characteristics of these new beats provided very strong evidence that they arose as a manifestation of IVR. In particular, the beats were shown to have phases and modulation depths dependent on the fluorescence band detected. Such behavior, which... 

The observation of novel quantum beats in the spectrally resolved fluorescence of anthracene21 forced one to consider, within the context of radiationless transition theory, the details of how IVR might be manifested in beat-modulated fluorescence decays. This work led to the concepts of phase-shifted quantum beats and restricted IVR,30a,4° and to a general set of results306 pertaining to the decays of spectrally resolved fluorescence in situations where an arbitrary number of vibrational levels, coupled by anharmonic coupling, participate in IVR. Moreover, three regimes of IVR have been identified no IVR, restricted (or coherent) IVR, and dissipative IVR.42... 

One notes immediately that Ia(t) and Ik(t) are both beat-modulated at an angular frequency of col2. Yet, one also notes two aspects in which the decays are qualitatively different. First, the a-type decay is modulated by a cosine term with positive coefficient. In contrast, the b-type decay has a cosine term with negative coefficient. The beats in the b-type decay are phase-shifted 180° from those in the a-type decay. Second, the a-type decay is not, in general, 100% modulated [2a2/ 2 (1 - 2a2/ 2)]. The b-type decay, however, is always 100% modulated. The modulation depths of the beat-modulated decays of the different bands are different. 

The eigenvector matrix C can be determined from relative values of quantum beat modulation depths, together with conditions associated with the fact that C is orthonormal. The first step is to work with an a-type decay and to note that... 

At this point it is worth noting that there are relations associated with the orthogonality of C and with absolute quantum beat modulation depths that also can be used to find the elements of C. Clearly, from the above, one may not need these relations. However, the analysis we have presented presumes that a good deal of high-quality experimental data are available. This may not always be the case. When it is not, these extra relations may be particularly useful. They are useful, in any case, in providing consistency checks on the elements calculated for the matrix. 

Decays of the spectrally resolved fluorescence from other S, levels in the low-energy regime of anthracene have also been measured. No beat-modulated, nonexponential decays have been observed, even with 80 psec temporal resolution. Both spectral and decay results indicate, therefore, that vibrational... 

Finally, Fig. 10 shows the decays of a third group of fluorescence bands in the 61 spectrum. It is apparent from the figure that all four bands decay in a similar manner. Fourier analysis of the decays confirms that this is indeed so. Figure 11 shows the Fourier spectrum of the decay at the top of Fig. 10. One notes that the same three beat components that are present here are present in the other two decay-types. Moreover, one notes (1) two — 1 beat phases and one +1 phase, (2) that the phase behavior in Fig. 11 is different from that in Fig. 9, and (3) that the sum of beat modulation depths is —0.70. 

Similar to the 61-level case, the time-resolved results pertaining to 51 excitation reveal fluorescence decays that are beat modulated and that depend on detection wavelength. Again, it is useful to consider groups of decays separately. 

Other excitation energies Other than the ones at S, + 1380 and S, + 1420 cm-, there are three prominent bands in the intermediate region of jet-cooled anthracene s excitation spectrum. Time- and frequency-resolved measurements subsequent to excitation of these bands have also been made. Without going into any detail concerning the results of these measurements, we do note that all three excitations give rise to quantum beat-modulated decays whose beat patterns (phases and modulation depths) depend on the fluorescence band detected.42 Figure 16 shows an example of this behavior for excitation to S, + 1514 cm-1. The two decays in the figure correspond to the detection of two different fluorescence bands in the S, + 1514 cm-1 fluorescence spectrum. 

Figure 24. Dispersed fluorescence spectra of jet-cooled t-stilbene for excitation energies that have been observed to give rise to beat-modulated fluorescence decays. Excess energies in S, for the excitations are given in cm-1 in the figure and the excitation wavelengths are marked with arrows in the spectra. The asterisks refer to detection wavelengths for the decays of Fig. 26. All of the spectra were obtained using similar experimental conditions that is, R = 0.5 A for most spectra, except R = 0.6S A for the vib = 821 and 860 cm-1 spectra and R = 0.32 A for the vib = 789 cm-1 spectrum.

In contrast to other vibrational levels in its vicinity, excitation to the level at E ib = 663 cm-1 gives rise to fluorescence bands, the decays of which are beat modulated. This is evident from Fig. 26 lower left, which shows the decay of the va = 800 cm-1 band in the vib = 663 cm-1 spectrum. (The spectrum appears in Fig. 24 lower left.) Fourier analysis reveals that this decay is modulated by a 780-MHz beat component having a +1 phase. Significantly, at least one other band in the same spectrum (va = 585 cm-1) is modulated at 780 MHz, but with the beat component having a — 1 phase. Therefore, unlike levels of similar energy, it appears that the t-stilbene level undergoes restricted IVR. 

Figure 26. Quantum beat-modulated fluorescence decays observed for excitation of various bands (the excess S, vibrational energies are given in cm-1 in the figure) of jet-cooled c-stilbene. The particular fluorescence band detected for each decay is given by an asterisk in the appropriate spectrum in Fig. 24. All decays were obtained with 80 psec temporal resolution except the ones corresponding to the S, + 852 and 860 cm-1 excitations, which were measured with 300 psec resolution. R for the decays was 1.6 A, except the S, +821 and 987 cm 1 decays for which R = 3.2 and 16.0 A, respectively.

Given the beat-modulated decays and phase-shift behavior, it is apparent that the intermediate-energy regime we have defined for S, t-stilbene is one in which restricted IVR is prevalent. Nevertheless, it is also apparent that IVR in f-stilbene does not fit as cleanly into the theory of vibrational coherence as anthracene does. This point will be addressed in subsection d. 

As previously discussed, if two or more excited eigenstates can combine in absorption with a common ground-state level, then these eigenstates can be excited so as to form a coherent superposition state. The superposition state, in turn, can give rise to quantum beat-modulated fluorescence decays. All this, of course, lies at the heart of the theory of vibrational coherence effects. However, it also implies that the same experimental conditions under which vibrational coherence effects are observed should allow for the observation of rotational coherence effects. That is, since more than one rotational level in the manifold of an excited vibronic state can combine in absorption with a single ground-state ro-vibrational level, then in a picosecond-resolved fluorescence experiment rotational quantum beats should obtain. 

The quantum beat modulation is strongest for tjs = tt/o) and the modulation is completely suppressed for ti3 = Itt/o). Surface trapping processes have to be additionally included in this model. With the decay of the population grating measured as S ti2 = 0, tx3) both Tx and T2 as well as the vibrational damping can be obtained from the mode-suppressed signal [302]. 

The bifurcation of a separatrix loop of a saddle-node was discovered by Andronov and Vitt [14] in their study of the transition phenomena from synchronization to beating modulations in radio-engineering. Specifically, they had studied the periodically forced van der Pol equation... 

In terms of the original variable (p — — ut, the stationary value of (the equilibrium state of system (12.1.9)) corresponds to an oscillatory regime with the same frequency as that of the external force. The periodic oscillations of (the limit cycle in (12.1.9)) correspond to a two-frequency regime. Hence, the above bifurcation scenario of a limit cycle from a homoclinic loop to a saddle-node characterizes the corresponding route from synchronization to beat modulations in Eq. (12.1.7). 

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